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Hiérarchies des modèles fluides M. Ottaviani (Ass. EURATOM-CEA, DRFC, Cadarache). Plan de l’exposé Rappel: échelles spatiales et temporelles Développement de basse fréquence 1: des équations de Braginskii aux modèles fluides de basse fréquence - PowerPoint PPT Presentation
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Hiérarchies des modèles fluides
M. Ottaviani
(Ass. EURATOM-CEA, DRFC, Cadarache)
Plan de l’exposé
• Rappel: échelles spatiales et temporelles
• Développement de basse fréquence 1: des équations de Braginskii aux modèles fluides de basse fréquence
• Développement de basse fréquence 2: de l’équation girocinétique aux modèles girofluides
• Exemples (issus de mes travaux, anciens ou en cours)
• Conclusions
Space and time scales in magnetic confinement
Low frequency dynamics: scales of interest
epea i c/
ITG
ETG
TEM
short- wave length ITG
plasma profile
zonal flows
zonal flows
E
a/vTe
a/vTi
Time
Space (r)
three scales
global
blobs
MHD
macro micromeso
From kinetic to fluid: classical approach
• Kinetic equation (collision dominated) ==>
• Chapman-Enskog expansion ==>
• Braginskii equations (1960) ==>
• low frequency expansions (drift ordering) ==>
• models, eventually ad hoc closures
From kinetic to fluid: gyrofluid approach (since 1990)
• Kinetic equation, non collisional (Vlasov) ==>
• low frequency expansion ==>
• gyrokinetic equation ==>
• moments, closures ==>
• hierarchy of gyrofluid models
Braginskii equations
Drift expansion
Generalized Ohm’s law
Hasegawa Mima model (1978)
Hasegawa-Wakatani model (1982)
Four field model, Hazeltine et al (1985)
Problems
Gyrofluid approach
Moments of the GKE (m+n models), Dorland (1992)
The issue of GF closures
Further developments
The 3+1 model from Beer (1994)
Hasegawa-Mima, revisited
Example 1. Collisionless reconnection: sawtooth crash
Result (Ottaviani & Porcelli, 1993)Current sheet formation on a fast timescale, controlled by el. inertia.
Example 2. Tearing-modes at low collisionality
• Analysis of tearing modes (reconnecting modes) in regimes of low collisionality. The drift frequency exceeds the collision frequency. Electron inertia comes into play
• Employs the four-field model
• Study linear stability criteria.
• Search for bistability (coexistence of states).
• Coupling to drift waves
Electromagnetic solution, islandlocalized electric potential
Electrostatic solution, small islandde-localized electric potential, drift-wave
Example 3. Ion temperature gradient turbulence
• General goal: determine the dependence of turbulent thermal transport as a function of dimensionless parameters (Manfredi & Ottaviani, 1997, and foll.)
• ITG model:
Conclusions (personal)
• Conventional low frequency fluid models still useful, at least for a first approach to a given problem
• Gyrofluid models more flexibles, take naturally into account the anisotropy
• Good practical closures for the parallel dynamics exist, if sufficiently high order momenta are kept, especially when magnetic fluctiations are present
• FLR closures still involved, complicated. Problematic at very short wavelengths (below the ion Larmor radius)
• More work ?
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